Page 159. " I would not venture over a horsepond in it. "
Horse Pond
Creative Commons Attribution Share AlikeHorse Pond - Credit: Simon Carey

A horsepond is a small body of water for the watering of horses, or a pond about the size that would serve that function.  In sailing terms, this is a label denoting insignificance.

 

Page 164. " she, Anne, and Mrs. Clay, therefore, turned into Molland's "

Molland's was a shop on Milsom Street in Bath where confections could be purchased and enjoyed.  Anne and her party briefly stop at the shop to avoid the rain. 

To this day, Milsom Street remains the main shopping street in Bath.

Milsom Street, Bath
Creative Commons Attribution Share AlikeMilsom Street, Bath - Credit: Donnylad

  

Page 164. " Her ladyship's carriage was a barouche, and did not hold more than four with any comfort. "

A type of carriage, the barouche was a stately horse-drawn vehicle with a top that could be taken down or put up depending on the weather or moods of the occupants. 

Barouche
Creative Commons Attribution Share AlikeBarouche - Credit: Pelikana

 

Typically seating four people, on two seats facing each other, a barouche like Lady Dalrymple's would have been an essential form of fashionable and elegant transportation.

Page 166. " I think, it would be more prudent to let me get you a chair. "

Captain Wentworth shows concern for Anne Elliot's umbrellaless condition on a rainy Bath day by encouraging her to use a sedan chair as a means of returning home. Sedan chairs were, as illustrated, a human-powered vehicle composed of an enclosed box and two poles. Decoration varied (this illustration is of a rather elaborate design for Queen Charlotte). 

 

Sedan Chair
Public DomainSedan Chair - Credit: Robert Adam

 

The poles would be manned, quite literally, and the chair's occupant would then be carried to his or her destination in relative comfort, if not speed.

Page 169. " The theatre or the rooms, where he was most likely to be, were not fashionable enough for the Elliots, whose evening amusements were solely in the elegant stupidity of private parties "
Royal Theatre
Creative Commons Attribution Share AlikeRoyal Theatre - Credit: Christophe Finot

The theatre is most likely the Theatre Royal and the rooms referred to here are the Assembly rooms, of which the octagon room (below) is one. The former was for plays and theatricals, the latter for balls, parties, and social elbow-rubbing.

 

Fashion Museum and Assembly Rooms
Creative Commons Attribution Share AlikeFashion Museum and Assembly Rooms - Credit: Mark Anderson

 The vanity and social ambition of Miss Elliot and Sir Walter are again revealed in their disdain of the forms of entertainment which are good enough for gentlemen like Captain Wentworth and the Crofts.

Both are still lively venues today. For more about the Theatre Royal, try their official site

Page 171. " they took their station by one of the fires in the octagon room "

The octagon room was one of the assembly (or Upper) rooms for society and entertainment in Regency Bath. 

Octagon Room, Bath
Creative Commons Attribution Share AlikeOctagon Room, Bath - Credit: Stephen McKay

The room from which the other assembly rooms branched out, it was often used for the same purpose -- a concert -- that brought Anne Elliot and her party together in this context.  These rooms are still used for similar purposes today.